Since \(\begin{bmatrix}-\mathbf {X}_{\mathbf {Q}} & \mathbf {Y}_{\mathbf {Q}}\end{bmatrix}\)  is stable and must satisfy the identity from (REF ), then we have that \(\begin{bmatrix}-\mathbf {X}_{\mathbf {Q}}(\lambda ) & \mathbf {Y}_{\mathbf {Q}}(\lambda )\end{bmatrix}\)  must have only finite entries along with full row rank \(\forall \lambda \in \mathbb {C}\backslash \mathbb {S}\) . This, in turn, means that \(\begin{bmatrix}\mathbf {Y}_{\mathbf {Q}} & -\mathbf {X}_{\mathbf {Q}}\end{bmatrix}\)  and every \(e_i^\top \begin{bmatrix}\mathbf {Y}_{\mathbf {Q}} & -\mathbf {X}_{\mathbf {Q}}\end{bmatrix}\)  \(\forall i\in 1:m\)  will share this property. Then, it follows that \(\begin{bmatrix}\mathbf {Y}_{\mathbf {Q}} & -\mathbf {X}_{\mathbf {Q}}\end{bmatrix}\)  and every \(e_i^\top \begin{bmatrix}\mathbf {Y}_{\mathbf {Q}} & -\mathbf {X}_{\mathbf {Q}}\end{bmatrix}\)  \(\forall i\in 1:m\)  are stable and that they have full row normal rank along with, by Lemma 3.29 in [1]}, no transmission zeros in \(\mathbb {C}\backslash \mathbb {S}\) .
